Question
Brandy is asked to sketch a graph of m(x)=(2x+5)(x-3)^3(8x+1)^5
. How many turning points should her sketch include?
A
at most 2
B
at most 7
C
at most 8
D
at most 10
Answers
GPT-4o mini
To determine the number of turning points for the function \( m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 \), we first need to analyze the factors in the polynomial.
1. **Identify the degree of the polynomial**: The degree of a polynomial gives an upper bound on the maximum number of turning points, which is at most \( n - 1 \), where \( n \) is the degree of the polynomial.
- The term \( (2x + 5) \) contributes a degree of 1.
- The term \( (x - 3)^3 \) contributes a degree of 3.
- The term \( (8x + 1)^5 \) contributes a degree of 5.
Thus, the total degree of \( m(x) \) is:
\[
1 + 3 + 5 = 9.
\]
2. **Turning points**: The number of turning points of a polynomial is at most \( n - 1 \):
\[
\text{Maximum turning points} = 9 - 1 = 8.
\]
So, Brandy should include at most **8 turning points** in her sketch.
Therefore, the correct answer is **C: at most 8**.
1. **Identify the degree of the polynomial**: The degree of a polynomial gives an upper bound on the maximum number of turning points, which is at most \( n - 1 \), where \( n \) is the degree of the polynomial.
- The term \( (2x + 5) \) contributes a degree of 1.
- The term \( (x - 3)^3 \) contributes a degree of 3.
- The term \( (8x + 1)^5 \) contributes a degree of 5.
Thus, the total degree of \( m(x) \) is:
\[
1 + 3 + 5 = 9.
\]
2. **Turning points**: The number of turning points of a polynomial is at most \( n - 1 \):
\[
\text{Maximum turning points} = 9 - 1 = 8.
\]
So, Brandy should include at most **8 turning points** in her sketch.
Therefore, the correct answer is **C: at most 8**.